In my mind, Frodo and Bilbo reach Valinor and live to whatever end of days they desire to endure as long as they stay there. For if the capricious Valar treat those who were "meant to carry the ring" like so much hired help who sit at a separate table for meals by not letting them onto Valinor proper, or they treat them as disposable by not "bestowing" immortality (I prefer to believe immortality is intrinsic to the land and not the Valar who bestow it), then the Valar are truly snobbish ingrates who do not deserve my respect for treating their agents against Sauron in such a poor manner.